There are two other transformations,
but they're harder to "see" with any degree of accuracy. If
you compare the graphs of2x2, x2,
and( 1/2 )x2,
you'll see what I mean:

2x2

x2

_1/2x2_

The parabola for2x2 grows twice as fast asx2,
so its graph is tall and skinny. On the other hand, the parabola for the
function( 1/2 )x2 grows only half as fast, so its graph is short and fat. You can tell,
roughly speaking, that the first graph is multiplied by something bigger
than1 and that the third graph is multiplied by something smaller than1.
But it is generally difficult to tell exactly what a graph has been multiplied
by, just by looking at the picture.

As you can see, multiplying
inside the function (inside the argument of the function) causes the graph
to get thinner or fatter. This looks a lot like the other multiplication
transformation, and is about impossible to identify from a graph. It helps
to look at the zeroes of the graph (if it has more than one). For instance,
looking at y = x2 – 4,
you can see that multiplying outside the function doesn't change the location
of the zeroes, but multiplying inside the function does:

2(x2 – 4)

x2 – 4

(2x)2 – 4

So the "left",
"right", "up", "down", "flip",
and "mirror" transformations are fairly straightforward, but
the "multiply" transformations, also called "stretching"
and "squeezing", can get a little messy. Hope that they aren't
frequently required of you.

Typical homework problems
on this topic ask you to graph the transformation of a function, given
the original function, or else ask you to figure out the transformation,
given the comparative graphs.

Thinking of the graph
off(x)
= x4,
graphf(x – 2) + 1

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The graph of f(x) looks like this:

Looking at the expression
for the transformation, the "+1"
outside tells me that the graph is going to be moved up by one unit.
And the "–2"
inside the argument tells me that the graph is going to be shifted two
units RIGHT. (Remember that the left-right shifting is backwards from
what you might expect.)

Then my graph looks
like this:

When they are having you
graph by moving other graphs around, they can't be terribly critical of
your drawing, since you're not supposed to be making a T-chart and computing
exact points. But do try to make your graph look reasonable.

You can always "cheat",
by the way, especially if you have a graphing calculator, by quickly graphing(x – 2)4 + 1and verifying that
it matches what you've drawn. But you do need to know how to do function
transformations, because there are ways to ask the questions that don't
allow you to cheat....